LMFDB - Embedded newform 2793.1.eo.a.1187.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2793,1,Mod(137,2793)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2793, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 102, 14]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2793.137");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2793.eo (of order $126$, degree $36$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.39388858028$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Embedding invariants

Embedding label			1187.1
Root			$-0.318487 + 0.947927i$ of defining polynomial
Character	$\chi$	$=$	2793.1187
Dual form			2793.1.eo.a.2753.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.661686 + 0.749781i) q^{3} +(0.270840 + 0.962624i) q^{4} +(-0.797133 - 0.603804i) q^{7} +(-0.124344 - 0.992239i) q^{9} +O(q^{10})$	\(q+(-0.661686 + 0.749781i) q^{3} +(0.270840 + 0.962624i) q^{4} +(-0.797133 - 0.603804i) q^{7} +(-0.124344 - 0.992239i) q^{9} +(-0.900969 - 0.433884i) q^{12} +(0.801308 - 1.56301i) q^{13} +(-0.853291 + 0.521435i) q^{16} +(-0.733052 - 0.680173i) q^{19} +(0.980172 - 0.198146i) q^{21} +(-0.124344 - 0.992239i) q^{25} +(0.826239 + 0.563320i) q^{27} +(0.365341 - 0.930874i) q^{28} +(-0.995031 - 1.72344i) q^{31} +(0.921476 - 0.388435i) q^{36} +(-0.0614710 - 0.820274i) q^{37} +(0.641701 + 1.63503i) q^{39} +(-0.778561 + 0.475769i) q^{43} +(0.173648 - 0.984808i) q^{48} +(0.270840 + 0.962624i) q^{49} +(1.72162 + 0.348032i) q^{52} +(0.995031 - 0.0995678i) q^{57} +(-0.818487 - 1.81395i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.733052 - 0.680173i) q^{64} +(1.34551 + 1.12902i) q^{67} +(1.16604 + 0.0581944i) q^{73} +(0.826239 + 0.563320i) q^{75} +(0.456211 - 0.889872i) q^{76} +(-1.31226 - 0.477622i) q^{79} +(-0.969077 + 0.246757i) q^{81} +(0.456211 + 0.889872i) q^{84} +(-1.58250 + 0.762092i) q^{91} +(1.95060 + 0.394323i) q^{93} +(0.331867 + 1.88211i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * q	$ 36 q - 6 q^{12} + 3 q^{13} + 3 q^{19} + 3 q^{27} + 3 q^{28} + 3 q^{43} + 3 q^{52} - 18 q^{61} - 18 q^{63} + 3 q^{64} + 3 q^{67} - 6 q^{73} + 3 q^{75} - 6 q^{79} + 3 q^{93}+O(q^{100}) $ 36 * q - 6 * q^12 + 3 * q^13 + 3 * q^19 + 3 * q^27 + 3 * q^28 + 3 * q^43 + 3 * q^52 - 18 * q^61 - 18 * q^63 + 3 * q^64 + 3 * q^67 - 6 * q^73 + 3 * q^75 - 6 * q^79 + 3 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times$.

$n$	$932$	$2110$	$2206$
$\chi(n)$	$-1$	$e\left(\frac{20}{21}\right)$	$e\left(\frac{4}{9}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$2$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$3$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$3$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$4$	0.270840	+	0.962624i	0.270840	+	0.962624i
$5$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$5$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$6$		0			0
$7$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$7$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$8$		0			0
$9$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$10$		0			0
$11$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$11$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$12$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$13$	0.801308	−	1.56301i	0.801308	−	1.56301i	−0.0249307	−	0.999689i	$-0.507937\pi$
$13$	0.801308	−	1.56301i	0.801308	−	1.56301i	0.826239	−	0.563320i	$-0.190476\pi$
$14$		0			0
$15$		0			0
$16$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$17$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$17$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$18$		0			0
$19$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$19$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$20$		0			0
$21$	0.980172	−	0.198146i	0.980172	−	0.198146i
$22$		0			0
$23$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$23$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$24$		0			0
$25$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$26$		0			0
$27$	0.826239	+	0.563320i	0.826239	+	0.563320i
$28$	0.365341	−	0.930874i	0.365341	−	0.930874i
$29$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$29$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$30$		0			0
$31$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.583744	−	0.811938i	$-0.698413\pi$
$31$	−0.995031	−	1.72344i	−0.995031	−	1.72344i	−0.411287	−	0.911506i	$-0.634921\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.921476	−	0.388435i	0.921476	−	0.388435i
$37$	−0.0614710	−	0.820274i	−0.0614710	−	0.820274i	−0.939693	−	0.342020i	$-0.888889\pi$
$37$	−0.0614710	−	0.820274i	−0.0614710	−	0.820274i	0.878222	−	0.478254i	$-0.158730\pi$
$38$		0			0
$39$	0.641701	+	1.63503i	0.641701	+	1.63503i
$40$		0			0
$41$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$41$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$42$		0			0
$43$	−0.778561	+	0.475769i	−0.778561	+	0.475769i	−0.853291	−	0.521435i	$-0.825397\pi$
$43$	−0.778561	+	0.475769i	−0.778561	+	0.475769i	0.0747301	+	0.997204i	$0.476190\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$47$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$48$	0.173648	−	0.984808i	0.173648	−	0.984808i
$49$	0.270840	+	0.962624i	0.270840	+	0.962624i
$50$		0			0
$51$		0			0
$52$	1.72162	+	0.348032i	1.72162	+	0.348032i
$53$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$53$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$58$		0			0
$59$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$59$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$60$		0			0
$61$	−0.818487	−	1.81395i	−0.818487	−	1.81395i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−0.818487	−	1.81395i	−0.818487	−	1.81395i	−0.318487	−	0.947927i	$-0.603175\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$64$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$65$		0			0
$66$		0			0
$67$	1.34551	+	1.12902i	1.34551	+	1.12902i	0.980172	+	0.198146i	$0.0634921\pi$
$67$	1.34551	+	1.12902i	1.34551	+	1.12902i	0.365341	+	0.930874i	$0.380952\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$71$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$72$		0			0
$73$	1.16604	+	0.0581944i	1.16604	+	0.0581944i	0.623490	−	0.781831i	$-0.285714\pi$
$73$	1.16604	+	0.0581944i	1.16604	+	0.0581944i	0.542546	+	0.840026i	$0.317460\pi$
$74$		0			0
$75$	0.826239	+	0.563320i	0.826239	+	0.563320i
$76$	0.456211	−	0.889872i	0.456211	−	0.889872i
$77$		0			0
$78$		0			0
$79$	−1.31226	−	0.477622i	−1.31226	−	0.477622i	−0.411287	−	0.911506i	$-0.634921\pi$
$79$	−1.31226	−	0.477622i	−1.31226	−	0.477622i	−0.900969	+	0.433884i	$0.857143\pi$
$80$		0			0
$81$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$82$		0			0
$83$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$83$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$84$	0.456211	+	0.889872i	0.456211	+	0.889872i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.921476	−	0.388435i	$-0.873016\pi$
$89$		0			0		0.921476	+	0.388435i	$0.126984\pi$
$90$		0			0
$91$	−1.58250	+	0.762092i	−1.58250	+	0.762092i
$92$		0			0
$93$	1.95060	+	0.394323i	1.95060	+	0.394323i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.331867	+	1.88211i	0.331867	+	1.88211i	0.456211	+	0.889872i	$0.349206\pi$
$97$	0.331867	+	1.88211i	0.331867	+	1.88211i	−0.124344	+	0.992239i	$0.539683\pi$
$98$		0			0
$99$		0			0
$100$	0.921476	−	0.388435i	0.921476	−	0.388435i
$101$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$101$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$102$		0			0
$103$	1.57646	−	0.237613i	1.57646	−	0.237613i	0.698237	−	0.715867i	$-0.253968\pi$
$103$	1.57646	−	0.237613i	1.57646	−	0.237613i	0.878222	+	0.478254i	$0.158730\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$107$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$108$	−0.318487	+	0.947927i	−0.318487	+	0.947927i
$109$	1.57646	+	1.19412i	1.57646	+	1.19412i	0.878222	+	0.478254i	$0.158730\pi$
$109$	1.57646	+	1.19412i	1.57646	+	1.19412i	0.698237	+	0.715867i	$0.253968\pi$
$110$		0			0
$111$	0.655701	+	0.496674i	0.655701	+	0.496674i
$112$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$113$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$113$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.65052	−	0.600739i	−1.65052	−	0.600739i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$122$		0			0
$123$		0			0
$124$	1.38953	−	1.42462i	1.38953	−	1.42462i
$125$		0			0
$126$		0			0
$127$	0.183040	+	0.405658i	0.183040	+	0.405658i	0.980172	−	0.198146i	$-0.0634921\pi$
$127$	0.183040	+	0.405658i	0.183040	+	0.405658i	−0.797133	+	0.603804i	$0.793651\pi$
$128$		0			0
$129$	0.158440	−	0.898560i	0.158440	−	0.898560i
$130$		0			0
$131$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$131$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$132$		0			0
$133$	0.173648	+	0.984808i	0.173648	+	0.984808i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.661686	−	0.749781i	$-0.730159\pi$
$137$		0			0		0.661686	+	0.749781i	$0.269841\pi$
$138$		0			0
$139$	0.00124308	−	0.0498459i	0.00124308	−	0.0498459i	−0.998757	−	0.0498459i	$-0.984127\pi$
$139$	0.00124308	−	0.0498459i	0.00124308	−	0.0498459i	1.00000			$0$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.623490	+	0.781831i	0.623490	+	0.781831i
$145$		0			0
$146$		0			0
$147$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$148$	0.772967	−	0.281337i	0.772967	−	0.281337i
$149$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$149$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$150$		0			0
$151$	−0.109562	+	1.46200i	−0.109562	+	1.46200i	0.623490	+	0.781831i	$0.285714\pi$
$151$	−0.109562	+	1.46200i	−0.109562	+	1.46200i	−0.733052	+	0.680173i	$0.761905\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.40012	+	1.06055i	−1.40012	+	1.06055i
$157$	−0.0488728	−	1.95974i	−0.0488728	−	1.95974i	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	−0.0488728	−	1.95974i	−0.0488728	−	1.95974i	0.173648	−	0.984808i	$-0.444444\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.232712	−	0.592942i	−0.232712	−	0.592942i	0.766044	−	0.642788i	$-0.222222\pi$
$163$	−0.232712	−	0.592942i	−0.232712	−	0.592942i	−0.998757	+	0.0498459i	$0.984127\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$167$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$168$		0			0
$169$	−1.21716	−	1.69297i	−1.21716	−	1.69297i
$170$		0			0
$171$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$172$	−0.668852	−	0.620604i	−0.668852	−	0.620604i
$173$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$173$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$179$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$180$		0			0
$181$	−1.82237	−	0.768193i	−1.82237	−	0.768193i	−0.969077	−	0.246757i	$-0.920635\pi$
$181$	−1.82237	−	0.768193i	−1.82237	−	0.768193i	−0.853291	−	0.521435i	$-0.825397\pi$
$182$		0			0
$183$	1.90165	+	0.586581i	1.90165	+	0.586581i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$190$		0			0
$191$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$191$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$192$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$193$	0.421476	−	0.477591i	0.421476	−	0.477591i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.421476	−	0.477591i	0.421476	−	0.477591i	0.921476	+	0.388435i	$0.126984\pi$
$194$		0			0
$195$		0			0
$196$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	0.0497994	−	1.99689i	0.0497994	−	1.99689i	−0.0249307	−	0.999689i	$-0.507937\pi$
$199$	0.0497994	−	1.99689i	0.0497994	−	1.99689i	0.0747301	−	0.997204i	$-0.476190\pi$
$200$		0			0
$201$	−1.73683	+	0.261784i	−1.73683	+	0.261784i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.131259	+	1.75153i	0.131259	+	1.75153i
$209$		0			0
$210$		0			0
$211$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.222521	−	0.974928i	$-0.571429\pi$
$211$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.318487	+	0.947927i	$0.603175\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.247452	+	1.97462i	−0.247452	+	1.97462i
$218$		0			0
$219$	−0.815183	+	0.835765i	−0.815183	+	0.835765i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.92852	+	0.192978i	−1.92852	+	0.192978i	−0.988831	−	0.149042i	$-0.952381\pi$
$223$	−1.92852	+	0.192978i	−1.92852	+	0.192978i	−0.939693	+	0.342020i	$0.888889\pi$
$224$		0			0
$225$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	0.365341	+	0.930874i	0.365341	+	0.930874i
$229$	0.245910	+	0.0370649i	0.245910	+	0.0370649i	0.270840	−	0.962624i	$-0.412698\pi$
$229$	0.245910	+	0.0370649i	0.245910	+	0.0370649i	−0.0249307	+	0.999689i	$0.507937\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.270840	−	0.962624i	$-0.412698\pi$
$233$		0			0		−0.270840	+	0.962624i	$0.587302\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.22641	−	0.667869i	1.22641	−	0.667869i
$238$		0			0
$239$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$239$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$240$		0			0
$241$	1.36879	−	1.40335i	1.36879	−	1.40335i	0.542546	−	0.840026i	$-0.317460\pi$
$241$	1.36879	−	1.40335i	1.36879	−	1.40335i	0.826239	−	0.563320i	$-0.190476\pi$
$242$		0			0
$243$	0.456211	−	0.889872i	0.456211	−	0.889872i
$244$	1.52448	−	1.27919i	1.52448	−	1.27919i
$245$		0			0
$246$		0			0
$247$	−1.65052	+	0.600739i	−1.65052	+	0.600739i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$251$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$252$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.456211	−	0.889872i	0.456211	−	0.889872i
$257$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$257$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$258$		0			0
$259$	−0.446285	+	0.690984i	−0.446285	+	0.690984i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$263$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.722402	+	1.60101i	−0.722402	+	1.60101i
$269$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$269$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$270$		0			0
$271$	−0.426531	−	0.593269i	−0.426531	−	0.593269i	0.542546	−	0.840026i	$-0.317460\pi$
$271$	−0.426531	−	0.593269i	−0.426531	−	0.593269i	−0.969077	+	0.246757i	$0.920635\pi$
$272$		0			0
$273$	0.475716	−	1.69079i	0.475716	−	1.69079i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	0.0747301	−	0.997204i	$-0.476190\pi$
$277$	−0.658322	−	0.317031i	−0.658322	−	0.317031i	−0.733052	+	0.680173i	$0.761905\pi$
$278$		0			0
$279$	−1.58634	+	1.20161i	−1.58634	+	1.20161i
$280$		0			0
$281$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$281$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$282$		0			0
$283$	0.224060	−	1.78795i	0.224060	−	1.78795i	−0.318487	−	0.947927i	$-0.603175\pi$
$283$	0.224060	−	1.78795i	0.224060	−	1.78795i	0.542546	−	0.840026i	$-0.317460\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$290$		0			0
$291$	−1.63076	−	0.996539i	−1.63076	−	0.996539i
$292$	0.259790	+	1.13822i	0.259790	+	1.13822i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.318487	+	0.947927i	−0.318487	+	0.947927i
$301$	0.907887	+	0.0908478i	0.907887	+	0.0908478i
$302$		0			0
$303$		0			0
$304$	0.980172	+	0.198146i	0.980172	+	0.198146i
$305$		0			0
$306$		0			0
$307$	1.16868	+	0.885240i	1.16868	+	0.885240i	0.995031	−	0.0995678i	$-0.0317460\pi$
$307$	1.16868	+	0.885240i	1.16868	+	0.885240i	0.173648	+	0.984808i	$0.444444\pi$
$308$		0			0
$309$	−0.864963	+	1.33922i	−0.864963	+	1.33922i
$310$		0			0
$311$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$311$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$312$		0			0
$313$	−1.55282	+	0.565181i	−1.55282	+	0.565181i	−0.969077	−	0.246757i	$-0.920635\pi$
$313$	−1.55282	+	0.565181i	−1.55282	+	0.565181i	−0.583744	+	0.811938i	$0.698413\pi$
$314$		0			0
$315$		0			0
$316$	0.104359	−	1.39257i	0.104359	−	1.39257i
$317$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$317$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$	−1.65052	−	0.600739i	−1.65052	−	0.600739i
$326$		0			0
$327$	−1.93845	+	0.391866i	−1.93845	+	0.391866i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.896546	−	0.611254i	0.896546	−	0.611254i	−0.0249307	−	0.999689i	$-0.507937\pi$
$331$	0.896546	−	0.611254i	0.896546	−	0.611254i	0.921476	+	0.388435i	$0.126984\pi$
$332$		0			0
$333$	−0.806265	+	0.162990i	−0.806265	+	0.162990i
$334$		0			0
$335$		0			0
$336$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$337$	1.12922	−	0.690053i	1.12922	−	0.690053i	0.173648	−	0.984808i	$-0.444444\pi$
$337$	1.12922	−	0.690053i	1.12922	−	0.690053i	0.955573	+	0.294755i	$0.0952381\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.365341	−	0.930874i	0.365341	−	0.930874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$347$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$348$		0			0
$349$	0.673306	+	1.71556i	0.673306	+	1.71556i	0.698237	+	0.715867i	$0.253968\pi$
$349$	0.673306	+	1.71556i	0.673306	+	1.71556i	−0.0249307	+	0.999689i	$0.507937\pi$
$350$		0			0
$351$	1.54255	−	0.840026i	1.54255	−	0.840026i
$352$		0			0
$353$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$353$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$359$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$360$		0			0
$361$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$362$		0			0
$363$	0.270840	−	0.962624i	0.270840	−	0.962624i
$364$	−1.16221	−	1.31695i	−1.16221	−	1.31695i
$365$		0			0
$366$		0			0
$367$	0.212203	+	1.69334i	0.212203	+	1.69334i	0.623490	+	0.781831i	$0.285714\pi$
$367$	0.212203	+	1.69334i	0.212203	+	1.69334i	−0.411287	+	0.911506i	$0.634921\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.148717	+	1.98450i	0.148717	+	1.98450i
$373$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$373$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.203033	−	0.889545i	−0.203033	−	0.889545i	−0.969077	−	0.246757i	$-0.920635\pi$
$379$	−0.203033	−	0.889545i	−0.203033	−	0.889545i	0.766044	−	0.642788i	$-0.222222\pi$
$380$		0			0
$381$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$382$		0			0
$383$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$383$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.568885	+	0.713360i	0.568885	+	0.713360i
$388$	−1.72188	+	0.829215i	−1.72188	+	0.829215i
$389$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$389$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.89973	−	0.384038i	−1.89973	−	0.384038i	−0.900969	−	0.433884i	$-0.857143\pi$
$397$	−1.89973	−	0.384038i	−1.89973	−	0.384038i	−0.998757	+	0.0498459i	$0.984127\pi$
$398$		0			0
$399$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$400$	0.623490	+	0.781831i	0.623490	+	0.781831i
$401$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$401$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$402$		0			0
$403$	−3.49108	+	0.174233i	−3.49108	+	0.174233i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.0614710	−	0.136234i	−0.0614710	−	0.136234i	0.878222	−	0.478254i	$-0.158730\pi$
$409$	−0.0614710	−	0.136234i	−0.0614710	−	0.136234i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$	0.655701	+	1.45318i	0.655701	+	1.45318i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.0365510	+	0.0339144i	0.0365510	+	0.0339144i
$418$		0			0
$419$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$419$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$420$		0			0
$421$	−1.20842	+	0.307701i	−1.20842	+	0.307701i	−0.797133	−	0.603804i	$-0.793651\pi$
$421$	−1.20842	+	0.307701i	−1.20842	+	0.307701i	−0.411287	+	0.911506i	$0.634921\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.442830	+	1.94017i	−0.442830	+	1.94017i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$431$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$432$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$433$	0.212203	+	0.129674i	0.212203	+	0.129674i	0.623490	−	0.781831i	$-0.285714\pi$
$433$	0.212203	+	0.129674i	0.212203	+	0.129674i	−0.411287	+	0.911506i	$0.634921\pi$
$434$		0			0
$435$		0			0
$436$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$437$		0			0
$438$		0			0
$439$	1.36037	+	1.03044i	1.36037	+	1.03044i	0.995031	+	0.0995678i	$0.0317460\pi$
$439$	1.36037	+	1.03044i	1.36037	+	1.03044i	0.365341	+	0.930874i	$0.380952\pi$
$440$		0			0
$441$	0.921476	−	0.388435i	0.921476	−	0.388435i
$442$		0			0
$443$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$443$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$444$	−0.300520	+	0.765713i	−0.300520	+	0.765713i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.173648	+	0.984808i	0.173648	+	0.984808i
$449$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$449$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.02369	−	1.04954i	−1.02369	−	1.04954i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.15445	+	0.174005i	1.15445	+	0.174005i	0.698237	−	0.715867i	$-0.253968\pi$
$457$	1.15445	+	0.174005i	1.15445	+	0.174005i	0.456211	+	0.889872i	$0.349206\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$461$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$462$		0			0
$463$	−0.00372615	+	0.0497220i	−0.00372615	+	0.0497220i	−0.998757	−	0.0498459i	$-0.984127\pi$
$463$	−0.00372615	+	0.0497220i	−0.00372615	+	0.0497220i	0.995031	+	0.0995678i	$0.0317460\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$467$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$468$	0.131259	−	1.75153i	0.131259	−	1.75153i
$469$	−0.390845	−	1.71241i	−0.390845	−	1.71241i
$470$		0			0
$471$	1.50171	+	1.26009i	1.50171	+	1.26009i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$479$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$480$		0			0
$481$	−1.33135	−	0.561212i	−1.33135	−	0.561212i
$482$		0			0
$483$		0			0
$484$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$485$		0			0
$486$		0			0
$487$	0.0546039	−	0.139129i	0.0546039	−	0.139129i	−0.900969	−	0.433884i	$-0.857143\pi$
$487$	0.0546039	−	0.139129i	0.0546039	−	0.139129i	0.955573	+	0.294755i	$0.0952381\pi$
$488$		0			0
$489$	0.598559	+	0.217858i	0.598559	+	0.217858i
$490$		0			0
$491$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$491$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.74771	+	0.951755i	1.74771	+	0.951755i
$497$		0			0
$498$		0			0
$499$	0.821552	+	0.0410019i	0.821552	+	0.0410019i	0.456211	−	0.889872i	$-0.349206\pi$
$499$	0.821552	+	0.0410019i	0.821552	+	0.0410019i	0.365341	+	0.930874i	$0.380952\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$503$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.07473	+	0.207608i	2.07473	+	0.207608i
$508$	−0.340922	+	0.286067i	−0.340922	+	0.286067i
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$	−0.894347	−	0.750446i	−0.894347	−	0.750446i
$512$		0			0
$513$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$514$		0			0
$515$		0			0
$516$	0.907887	−	0.0908478i	0.907887	−	0.0908478i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.261979	−	0.779741i	0.261979	−	0.779741i	−0.733052	−	0.680173i	$-0.761905\pi$
$523$	0.261979	−	0.779741i	0.261979	−	0.779741i	0.995031	−	0.0995678i	$-0.0317460\pi$
$524$		0			0
$525$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.878222	+	0.478254i	0.878222	+	0.478254i
$530$		0			0
$531$		0			0
$532$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.08374	−	1.67796i	−1.08374	−	1.67796i	−0.583744	−	0.811938i	$-0.698413\pi$
$541$	−1.08374	−	1.67796i	−1.08374	−	1.67796i	−0.500000	−	0.866025i	$-0.666667\pi$
$542$		0			0
$543$	1.78181	−	0.858075i	1.78181	−	0.858075i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.65381	+	1.01062i	1.65381	+	1.01062i	0.955573	+	0.294755i	$0.0952381\pi$
$547$	1.65381	+	1.01062i	1.65381	+	1.01062i	0.698237	+	0.715867i	$0.253968\pi$
$548$		0			0
$549$	−1.69810	+	1.03769i	−1.69810	+	1.03769i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.757652	+	1.17307i	0.757652	+	1.17307i
$554$		0			0
$555$		0			0
$556$	0.0483195	−	0.0123037i	0.0483195	−	0.0123037i
$557$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$557$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$558$		0			0
$559$	0.119764	+	1.59813i	0.119764	+	1.59813i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$563$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.921476	+	0.388435i	0.921476	+	0.388435i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	1.15382	−	0.786662i	1.15382	−	0.786662i	0.173648	−	0.984808i	$-0.444444\pi$
$571$	1.15382	−	0.786662i	1.15382	−	0.786662i	0.980172	+	0.198146i	$0.0634921\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$577$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	−0.988831	−	0.149042i	$-0.952381\pi$
$577$	−0.914101	+	0.848162i	−0.914101	+	0.848162i	0.0747301	+	0.997204i	$0.476190\pi$
$578$		0			0
$579$	0.0792036	+	0.632030i	0.0792036	+	0.632030i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$587$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$588$	0.173648	−	0.984808i	0.173648	−	0.984808i
$589$	−0.442830	+	1.94017i	−0.442830	+	1.94017i
$590$		0			0
$591$		0			0
$592$	0.480172	+	0.667879i	0.480172	+	0.667879i
$593$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$593$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.46428	+	1.35865i	1.46428	+	1.35865i
$598$		0			0
$599$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$599$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$600$		0			0
$601$	1.46402	+	0.451591i	1.46402	+	0.451591i	0.921476	−	0.388435i	$-0.126984\pi$
$601$	1.46402	+	0.451591i	1.46402	+	0.451591i	0.542546	+	0.840026i	$0.317460\pi$
$602$		0			0
$603$	0.952952	−	1.47546i	0.952952	−	1.47546i
$604$	−1.43703	+	0.290503i	−1.43703	+	0.290503i
$605$		0			0
$606$		0			0
$607$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$607$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.0681853	+	0.133000i	0.0681853	+	0.133000i	0.921476	−	0.388435i	$-0.126984\pi$
$613$	0.0681853	+	0.133000i	0.0681853	+	0.133000i	−0.853291	+	0.521435i	$0.825397\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$617$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$618$		0			0
$619$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$619$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.40012	−	1.06055i	−1.40012	−	1.06055i
$625$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$626$		0			0
$627$		0			0
$628$	1.87325	−	0.577822i	1.87325	−	0.577822i
$629$		0			0
$630$		0			0
$631$	−0.172518	+	0.513474i	−0.172518	+	0.513474i	−0.998757	−	0.0498459i	$-0.984127\pi$
$631$	−0.172518	+	0.513474i	−0.172518	+	0.513474i	0.826239	+	0.563320i	$0.190476\pi$
$632$		0			0
$633$	0.378222	−	0.387771i	0.378222	−	0.387771i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.72162	+	0.348032i	1.72162	+	0.348032i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$641$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$642$		0			0
$643$	0.0329926	+	1.32296i	0.0329926	+	1.32296i	0.766044	+	0.642788i	$0.222222\pi$
$643$	0.0329926	+	1.32296i	0.0329926	+	1.32296i	−0.733052	+	0.680173i	$0.761905\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$647$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.31680	−	1.49211i	−1.31680	−	1.49211i
$652$	0.507752	−	0.384607i	0.507752	−	0.384607i
$653$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$653$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.0872464	−	1.16422i	−0.0872464	−	1.16422i
$658$		0			0
$659$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$659$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$660$		0			0
$661$	−0.921476	−	0.388435i	−0.921476	−	0.388435i	−0.124344	−	0.992239i	$-0.539683\pi$
$661$	−0.921476	−	0.388435i	−0.921476	−	0.388435i	−0.797133	+	0.603804i	$0.793651\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.13139	−	1.57366i	1.13139	−	1.57366i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.25101	−	1.16077i	1.25101	−	1.16077i	0.270840	−	0.962624i	$-0.412698\pi$
$673$	1.25101	−	1.16077i	1.25101	−	1.16077i	0.980172	−	0.198146i	$-0.0634921\pi$
$674$		0			0
$675$	0.456211	−	0.889872i	0.456211	−	0.889872i
$676$	1.30003	−	1.63019i	1.30003	−	1.63019i
$677$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$677$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$678$		0			0
$679$	0.871885	−	1.70067i	0.871885	−	1.70067i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$683$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$684$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$685$		0			0
$686$		0			0
$687$	−0.190506	+	0.159853i	−0.190506	+	0.159853i
$688$	0.416256	−	0.811938i	0.416256	−	0.811938i
$689$		0			0
$690$		0			0
$691$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$691$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$701$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$701$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$702$		0			0
$703$	−0.512867	+	0.643114i	−0.512867	+	0.643114i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0673546	−	0.239393i	−0.0673546	−	0.239393i	0.921476	−	0.388435i	$-0.126984\pi$
$709$	−0.0673546	−	0.239393i	−0.0673546	−	0.239393i	−0.988831	+	0.149042i	$0.952381\pi$
$710$		0			0
$711$	−0.310745	+	1.36146i	−0.310745	+	1.36146i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$719$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$720$		0			0
$721$	−1.40012	−	0.762464i	−1.40012	−	0.762464i
$722$		0			0
$723$	0.146497	+	1.95486i	0.146497	+	1.95486i
$724$	0.245910	−	1.96231i	0.245910	−	1.96231i
$725$		0			0
$726$		0			0
$727$	−0.110609	−	0.329212i	−0.110609	−	0.329212i	0.878222	−	0.478254i	$-0.158730\pi$
$727$	−0.110609	−	0.329212i	−0.110609	−	0.329212i	−0.988831	+	0.149042i	$0.952381\pi$
$728$		0			0
$729$	0.365341	+	0.930874i	0.365341	+	0.930874i
$730$		0			0
$731$		0			0
$732$	−0.0496136	+	1.98944i	−0.0496136	+	1.98944i
$733$	1.82624	+	0.563320i	1.82624	+	0.563320i	1.00000			$0$
$733$	1.82624	+	0.563320i	1.82624	+	0.563320i	0.826239	+	0.563320i	$0.190476\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.69810	+	0.169921i	−1.69810	+	0.169921i	−0.900969	−	0.433884i	$-0.857143\pi$
$739$	−1.69810	+	0.169921i	−1.69810	+	0.169921i	−0.797133	+	0.603804i	$0.793651\pi$
$740$		0			0
$741$	0.641701	−	1.63503i	0.641701	−	1.63503i
$742$		0			0
$743$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$743$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.507752	−	1.51125i	0.507752	−	1.51125i	−0.318487	−	0.947927i	$-0.603175\pi$
$751$	0.507752	−	1.51125i	0.507752	−	1.51125i	0.826239	−	0.563320i	$-0.190476\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.826239	−	0.563320i	0.826239	−	0.563320i
$757$	0.544286	−	1.20626i	0.544286	−	1.20626i	−0.411287	−	0.911506i	$-0.634921\pi$
$757$	0.544286	−	1.20626i	0.544286	−	1.20626i	0.955573	−	0.294755i	$-0.0952381\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$761$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$762$		0			0
$763$	−0.535631	−	1.90375i	−0.535631	−	1.90375i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.365341	+	0.930874i	0.365341	+	0.930874i
$769$	1.41178	+	0.595117i	1.41178	+	0.595117i	0.955573	−	0.294755i	$-0.0952381\pi$
$769$	1.41178	+	0.595117i	1.41178	+	0.595117i	0.456211	+	0.889872i	$0.349206\pi$
$770$		0			0
$771$		0			0
$772$	0.573893	+	0.276372i	0.573893	+	0.276372i
$773$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$773$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$774$		0			0
$775$	−1.58634	+	1.20161i	−1.58634	+	1.20161i
$776$		0			0
$777$	−0.222786	−	0.791830i	−0.222786	−	0.791830i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$785$		0			0
$786$		0			0
$787$	0.870687	+	1.09181i	0.870687	+	1.09181i	0.995031	+	0.0995678i	$0.0317460\pi$
$787$	0.870687	+	1.09181i	0.870687	+	1.09181i	−0.124344	+	0.992239i	$0.539683\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.49108	−	0.174233i	−3.49108	−	0.174233i
$794$		0			0
$795$		0			0
$796$	1.93575	−	0.492901i	1.93575	−	0.492901i
$797$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$797$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.722402	−	1.60101i	−0.722402	−	1.60101i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$809$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$810$		0			0
$811$	−0.708087	−	0.180301i	−0.708087	−	0.180301i	−0.124344	−	0.992239i	$-0.539683\pi$
$811$	−0.708087	−	0.180301i	−0.708087	−	0.180301i	−0.583744	+	0.811938i	$0.698413\pi$
$812$		0			0
$813$	0.727051	+	0.0727524i	0.727051	+	0.0727524i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.894330	+	0.180793i	0.894330	+	0.180793i
$818$		0			0
$819$	0.952952	+	1.47546i	0.952952	+	1.47546i
$820$		0			0
$821$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$821$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$822$		0			0
$823$	0.466934	−	1.38976i	0.466934	−	1.38976i	−0.411287	−	0.911506i	$-0.634921\pi$
$823$	0.466934	−	1.38976i	0.466934	−	1.38976i	0.878222	−	0.478254i	$-0.158730\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.661686	−	0.749781i	$-0.730159\pi$
$827$		0			0		0.661686	+	0.749781i	$0.269841\pi$
$828$		0			0
$829$	1.15382	+	0.786662i	1.15382	+	0.786662i	0.980172	−	0.198146i	$-0.0634921\pi$
$829$	1.15382	+	0.786662i	1.15382	+	0.786662i	0.173648	+	0.984808i	$0.444444\pi$
$830$		0			0
$831$	0.673306	−	0.283822i	0.673306	−	0.283822i
$832$	−1.65052	+	0.600739i	−1.65052	+	0.600739i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.148717	−	1.98450i	0.148717	−	1.98450i
$838$		0			0
$839$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$839$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$840$		0			0
$841$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$842$		0			0
$843$		0			0
$844$	−0.120535	−	0.528100i	−0.120535	−	0.528100i
$845$		0			0
$846$		0			0
$847$	0.980172	+	0.198146i	0.980172	+	0.198146i
$848$		0			0
$849$	1.19232	+	1.35106i	1.19232	+	1.35106i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.247121	−	0.878319i	0.247121	−	0.878319i	−0.733052	−	0.680173i	$-0.761905\pi$
$853$	0.247121	−	0.878319i	0.247121	−	0.878319i	0.980172	−	0.198146i	$-0.0634921\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$857$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$858$		0			0
$859$	−0.884207	−	0.225147i	−0.884207	−	0.225147i	−0.222521	−	0.974928i	$-0.571429\pi$
$859$	−0.884207	−	0.225147i	−0.884207	−	0.225147i	−0.661686	+	0.749781i	$0.730159\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$868$	−1.96783	+	0.296603i	−1.96783	+	0.296603i
$869$		0			0
$870$		0			0
$871$	2.84284	−	1.19836i	2.84284	−	1.19836i
$872$		0			0
$873$	1.82624	−	0.563320i	1.82624	−	0.563320i
$874$		0			0
$875$		0			0
$876$	−1.02531	−	0.558355i	−1.02531	−	0.558355i
$877$	0.507752	+	1.51125i	0.507752	+	1.51125i	0.826239	+	0.563320i	$0.190476\pi$
$877$	0.507752	+	1.51125i	0.507752	+	1.51125i	−0.318487	+	0.947927i	$0.603175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−0.229801	+	1.30327i	−0.229801	+	1.30327i	0.623490	+	0.781831i	$0.285714\pi$
$883$	−0.229801	+	1.30327i	−0.229801	+	1.30327i	−0.853291	+	0.521435i	$0.825397\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$887$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$888$		0			0
$889$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$890$		0			0
$891$		0			0
$892$	−0.708087	−	1.80418i	−0.708087	−	1.80418i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−0.668852	+	0.620604i	−0.668852	+	0.620604i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.333345	+	0.650213i	0.333345	+	0.650213i	0.995031	−	0.0995678i	$-0.0317460\pi$
$907$	0.333345	+	0.650213i	0.333345	+	0.650213i	−0.661686	+	0.749781i	$0.730159\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$911$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$912$	−0.797133	+	0.603804i	−0.797133	+	0.603804i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0309227	+	0.246757i	0.0309227	+	0.246757i
$917$		0			0
$918$		0			0
$919$	1.53758	+	0.740458i	1.53758	+	0.740458i	0.995031	−	0.0995678i	$-0.0317460\pi$
$919$	1.53758	+	0.740458i	1.53758	+	0.740458i	0.542546	+	0.840026i	$0.317460\pi$
$920$		0			0
$921$	−1.43703	+	0.290503i	−1.43703	+	0.290503i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.806265	+	0.162990i	−0.806265	+	0.162990i
$926$		0			0
$927$	−0.431791	−	1.53468i	−0.431791	−	1.53468i
$928$		0			0
$929$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$929$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$930$		0			0
$931$	0.456211	−	0.889872i	0.456211	−	0.889872i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.06358	−	0.215007i	1.06358	−	0.215007i	0.365341	−	0.930874i	$-0.380952\pi$
$937$	1.06358	−	0.215007i	1.06358	−	0.215007i	0.698237	+	0.715867i	$0.253968\pi$
$938$		0			0
$939$	0.603718	−	1.53825i	0.603718	−	1.53825i
$940$		0			0
$941$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$941$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$947$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$948$	0.975069	+	0.999689i	0.975069	+	0.999689i
$949$	1.02531	−	1.77589i	1.02531	−	1.77589i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$953$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.48017	+	2.56373i	−1.48017	+	2.56373i
$962$		0			0
$963$		0			0
$964$	1.72162	+	0.937543i	1.72162	+	0.937543i
$965$		0			0
$966$		0			0
$967$	−0.229160	+	0.0965988i	−0.229160	+	0.0965988i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$	−0.229160	+	0.0965988i	−0.229160	+	0.0965988i	0.270840	+	0.962624i	$0.412698\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$971$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$972$	0.980172	+	0.198146i	0.980172	+	0.198146i
$973$	−0.0310881	+	0.0389832i	−0.0310881	+	0.0389832i
$974$		0			0
$975$	1.54255	−	0.840026i	1.54255	−	0.840026i
$976$	1.64427	+	1.12104i	1.64427	+	1.12104i
$977$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$977$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.988831	−	1.71271i	0.988831	−	1.71271i
$982$		0			0
$983$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$983$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.02531	−	1.42612i	−1.02531	−	1.42612i
$989$		0			0
$990$		0			0
$991$	1.69824	−	0.715867i	1.69824	−	0.715867i	0.698237	−	0.715867i	$-0.253968\pi$
$991$	1.69824	−	0.715867i	1.69824	−	0.715867i	1.00000			$0$
$992$		0			0
$993$	−0.134924	+	1.07667i	−0.134924	+	1.07667i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.630128	−	1.39651i	−0.630128	−	1.39651i	−0.900969	−	0.433884i	$-0.857143\pi$
$997$	−0.630128	−	1.39651i	−0.630128	−	1.39651i	0.270840	−	0.962624i	$-0.412698\pi$
$998$		0			0
$999$	0.411287	−	0.712370i	0.411287	−	0.712370i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.1.eo.a.1187.1	yes	36
3.2	odd	2	CM	2793.1.eo.a.1187.1	yes	36
19.17	even	9		2793.1.ei.a.2069.1	yes	36
49.9	even	21		2793.1.ei.a.1871.1	✓	36
57.17	odd	18		2793.1.ei.a.2069.1	yes	36
147.107	odd	42		2793.1.ei.a.1871.1	✓	36
931.891	even	63	inner	2793.1.eo.a.2753.1	yes	36
2793.2753	odd	126	inner	2793.1.eo.a.2753.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.1.ei.a.1871.1	✓	36	49.9	even	21
2793.1.ei.a.1871.1	✓	36	147.107	odd	42
2793.1.ei.a.2069.1	yes	36	19.17	even	9
2793.1.ei.a.2069.1	yes	36	57.17	odd	18
2793.1.eo.a.1187.1	yes	36	1.1	even	1	trivial
2793.1.eo.a.1187.1	yes	36	3.2	odd	2	CM
2793.1.eo.a.2753.1	yes	36	931.891	even	63	inner
2793.1.eo.a.2753.1	yes	36	2793.2753	odd	126	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2793.eo (of order \(126\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.661686 + 0.749781i) q^{3} +(0.270840 + 0.962624i) q^{4} +(-0.797133 - 0.603804i) q^{7} +(-0.124344 - 0.992239i) q^{9} +O(q^{10})\)	\(q+(-0.661686 + 0.749781i) q^{3} +(0.270840 + 0.962624i) q^{4} +(-0.797133 - 0.603804i) q^{7} +(-0.124344 - 0.992239i) q^{9} +(-0.900969 - 0.433884i) q^{12} +(0.801308 - 1.56301i) q^{13} +(-0.853291 + 0.521435i) q^{16} +(-0.733052 - 0.680173i) q^{19} +(0.980172 - 0.198146i) q^{21} +(-0.124344 - 0.992239i) q^{25} +(0.826239 + 0.563320i) q^{27} +(0.365341 - 0.930874i) q^{28} +(-0.995031 - 1.72344i) q^{31} +(0.921476 - 0.388435i) q^{36} +(-0.0614710 - 0.820274i) q^{37} +(0.641701 + 1.63503i) q^{39} +(-0.778561 + 0.475769i) q^{43} +(0.173648 - 0.984808i) q^{48} +(0.270840 + 0.962624i) q^{49} +(1.72162 + 0.348032i) q^{52} +(0.995031 - 0.0995678i) q^{57} +(-0.818487 - 1.81395i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.733052 - 0.680173i) q^{64} +(1.34551 + 1.12902i) q^{67} +(1.16604 + 0.0581944i) q^{73} +(0.826239 + 0.563320i) q^{75} +(0.456211 - 0.889872i) q^{76} +(-1.31226 - 0.477622i) q^{79} +(-0.969077 + 0.246757i) q^{81} +(0.456211 + 0.889872i) q^{84} +(-1.58250 + 0.762092i) q^{91} +(1.95060 + 0.394323i) q^{93} +(0.331867 + 1.88211i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * q	\( 36 q - 6 q^{12} + 3 q^{13} + 3 q^{19} + 3 q^{27} + 3 q^{28} + 3 q^{43} + 3 q^{52} - 18 q^{61} - 18 q^{63} + 3 q^{64} + 3 q^{67} - 6 q^{73} + 3 q^{75} - 6 q^{79} + 3 q^{93}+O(q^{100}) \) 36 * q - 6 * q^12 + 3 * q^13 + 3 * q^19 + 3 * q^27 + 3 * q^28 + 3 * q^43 + 3 * q^52 - 18 * q^61 - 18 * q^63 + 3 * q^64 + 3 * q^67 - 6 * q^73 + 3 * q^75 - 6 * q^79 + 3 * q^93

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